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Development of a Laser-Based Geometric Imperfection Measurement Platform

with Application to Cold-Formed Steel Construction

Article  in  Experimental Mechanics · July 2015

DOI: 10.1007/s11340-015-0072-7

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Development of a Laser-Based Geometric ImperfectionMeasurement Platform with Application to Cold-Formed SteelConstruction

X. Zhao1 & M. Tootkaboni2 & B.W. Schafer1

Received: 3 June 2015 /Accepted: 1 July 2015# Society for Experimental Mechanics 2015

Abstract Geometric imperfections play an important role inthe strength and behavior of thinwalled metal parts such asthose commonly used in cold-formed steel construction. Theobjective of this paper is to detail a newly developed imper-fection measurement platform, where the full-three dimen-sional (3D) imperfect geometry of a cold-formed steel mem-ber can be measured and reconstructed with reasonably highthroughput and sufficient accuracy to explore all relevant im-perfections. The measurement platform is composed of a two-dimensional laser sensor mounted on rotary and linear stages.Specimens are placed within a rotary ring and along a refer-ence beam. The linear stage provides a means to scan thelength of the specimen and the rotary stage a means to scanfrom any angle. Control of the custom linear and rotary stagesare detailed in the paper, and corresponding control accuraciesare studied. A procedure for registering the individual scansinto the complete 3D model reconstruction (point cloud) isprovided by specific calibration tests, from which rotationand translation quantities can be found to transform surfacesmeasured in local coordinates to a universal global coordinatesystem. A variety of different imperfection quantities, frombasic dimensions and out-of-plane deviations to moreadvanced spectral and modalbased imperfection magnitudes

may be rapidly processed from the 3D model. Accurateunderstanding of geometric imperfections is critical to thelong-term success of analysis-based design paradigms forcold-formed steel constructions. The apparatus detailed hereinwill be used to provide a broad suite of imperfection informa-tion to assist in these new efforts.

Keywords Non-contact measurement . Geometricimperfections . Cold-formed steel member

Introduction

Structural design requires that engineers be able to reliablypredict the strength of their creations. Typically, structuralengineering design employs ideal geometries—every memberis true and straight and produced at its exact nominal dimen-sions. This design convenience has long been recognized as afallacy [1], and structural engineering design has used a vari-ety of approaches to deal with this reality in design, and avariety of methods to try to measure the extent to which realmembers and structures deviate from their presumed formsand shapes. Load bearing cold-formed steel construction,whereby civil works such as entire buildings (Fig. 1(a)), ded-icated wall and roof systems, or storage racks are designedfrom members cold bent from sheet steel typically less than2 mm thick (Fig. 1(b)) may be particularly prone to initialgeometric deviations and may require unique design methodsto treat them. However, without detailed knowledge of actualimperfections one cannot advance our basic understanding ofsuch deviations and the design methods needed to take theminto account. The measurement platform detailed in this paperis specifically designed to produce the needed initial geomet-ric imperfection information for cold-formed steel members.

* X. Zhaoxzhao16@jhu.edu

M. Tootkabonimtootkaboni@umassd.edu

B. W. Schaferschafer@jhu.edu

1 Civil Engineering, Johns Hopkins University, Baltimore, MD, USA2 Civil and Environmental Engineering, University of Massachusetts

Dartmouth, Dartmounth, MA, USA

Experimental MechanicsDOI 10.1007/s11340-015-0072-7

Cold-formed steel members for civil building constructionare usually understood as assemblages of plate structures, i.e.,the shape is bent from a flat plate or coil and the resultinggeometry consists of nominally flat plates and sharp corners(Fig. 1(b)). This is in contrast to shell structures, which gen-erally have at least one degree of curvature throughout. Plateand shell structures are known to have different level of im-perfection sensitivity [2]; however, the large degree of elasticimperfection sensitivity for shells has resulted in much greaterstudy of geometric imperfections and imperfection measure-ment for these structures, thus a brief background on imper-fection studies for shell structures is provided here.

Intensive work on imperfection sensitivity in shell (largelyaeronautical) structures initiated in the 1940’s includingKarman’s seminal experimental work demonstrating the pow-erfully deleterious impact of geometric imperfections on shell

compressive capacity [3] and Koiter’s critical theoretical in-sight on the importance of imperfections coincident witheigenbuckling modes of the structure [4]. Large experimentalefforts, often with detailed imperfection measurement studiestypically using single point contact measurement devices(e.g., LVDTs) and placing the shells on rotisserie stages, con-tinued throughout the twentieth century. This notably includeda group centered at Caltech initiating in the 1970’s, e.g., [5]and a group at Delft initiating in the 1980’s e.g., [6] (see [7] forcomplete details) and resulted in databases for shell imperfec-tions [7] that are still updated today [8]. Modern shell mea-surement techniques are moving towards non-contactmethods (e.g., [9] is an early example). The advent of struc-tural design based on modern geometric and material nonlin-ear collapse shell finite element analysis (e.g., [10, 11]) ofstructures has re-awakened an interest in imperfection mea-surements in shell structures [12].

Strength of thin-walled cold-formed steel members doesnot exhibit the same drastic Bknockdown^ in elastic bucklingstrength that is observed in shells; however, important sensi-tivity to imperfections, 15 % peak strength variations for im-perfection magnitudes as small as 10 % of the thickness, areregularly observed. Moreover, the imperfection sensitivity isnot purely elastic and involves inelastic buckling and collapsemechanisms; thus strength predictions must include materialfactors (e.g., yield stress in addition to elastic buckling). Thefirst practical strength prediction for thin-walled plates wasprovided by von Karman [13], but was not useful for steeldesign until it was empirically corrected for the presence ofimperfections by Winter [14]. Today, strength formulas withempirical corrections for imperfections remain the dominantdesign method ([15, 16]).

Existing data on geometric imperfections for cold-formedsteel members was compiled in 1998 [17]. Measurements typ-ically included only a few sparse points employing simplisticout-of-straightness data gathered with straight edges and dialgauges, or calipers, or occasionally surveying theodolites. Anotable exception to this is the work of Young ([18, 19]) whoemployed a single point line laser in his studies to providelongitudinal imperfections for several segments of his testspecimens. More recently, limited classes of imperfections(global bow camber and twist) have been studied in detail[20] using a position transducer on a manual linear stage.The opportunity to use explicit understanding of imperfec-tions in geometric and material nonlinear shell finite elementanalysis is attractive for efficient cold-formed steel design, butmuch work remains to characterize the imperfections [21].

Conventional imperfection measurements utilizing singlepoint contact-measurement cannot easily collect full cross sec-tion information. For example, the corners of the cross sec-tions are a detail that is difficult to assess with these types ofmethods. In addition, the time consuming nature of currentsingle-point contact or non-contact methods has resulted in

Fig. 1 Typical cold-formed steel construction (a) load bearing floorsystem, walls also load bearing cold-formed steel (image courtesyClarkDietrich Building Systems), and (b) example of individual Zee-shapedmember as commonly used for purlins and girts inmetal buildings

Exp Mech

the database of known imperfections being relatively small.This impedes probabilistic assessments of the imperfectionsand their expected impact on strength. High fidelity imperfec-tion measurements are needed for capturing the full imperfectgeometry and high throughput methods are also needed forextending imperfection measurements out of the lab, and intothe manufacturing facilities for quality control. All of thesefactors lead to an interest in full-field non-contact measure-ment for cold-formed steel structural members.

Two candidate methods are actively being explored incold-formed steel research for full-field imperfectionmeasure-ment: photogrammetry and line lasers [22]. Photogrammetryis flexible, but has some challenges with respect to accuracy;thus, our group has focused on the application of 2D linelasers for imperfection geometry measurement [23]. The dis-advantage, of even a 2D line laser, is the need to run theinstrument over the specimen requiring at least some stagingand control in addition to merging of data to reconstruct themeasured 3D geometry. This paper details the laser-based im-perfection measurement platform we have designed and im-plemented. First, overall design is summarized, followed byexplanation of the linear and rotary stages for the laser. This isfollowed by an explanation of calibration and registration ofthe measurements in global coordinates, and an example.

Overall Rig Design

The objective of the imperfection measurement rig is toachieve reasonably high-throughput and high-accuracy repre-sentations of the three-dimensional geometry of as-manufactured members employed in cold-formed steel con-struction. To achieve sufficient accuracy a laser-based non-contact measurement system was selected. Given the desireto potentially adapt the system for specimens under load it wasdecided to place the laser sensor on a moveable stage, as

opposed to moving the specimen. Overall it was desired tobe able to measure a specimen of at least 250 mm in width ordepth and 3 m long; these constraints established the scanningarea. These dimensions are within typical dimensions used forwalls and floors in cold-formed steel construction. Theresulting imperfection measurement rig is shown in Fig. 2.The rig includes a laser scanner, a rotary stage, a triangularsupport frame and a linear motion system. These elements aredetailed in the following sections as well as briefly summa-rized here.

The laser scanner is a 2D line laser that generates 800points per reading, covering widths up to 240 mm. It is posi-tioned on the 635 mm diameter rotary stage. This allows thelaser scanner to scan different segments of a target specimen,e.g., corner radius in a cold-formed steel member. The supportframe where the rotary stage is attached is heavy and wideenough to carry the rotary ring without tipping and providesonly minor forces on the linear guides. Specimens are placedon a support beam to minimize deflections under gravityloads. The linear motion system drives the support frameand positions the laser along the length. Full-field geometricinformation of a target specimen can be achieved by scanningthe specimen at multiple angles of view and registering theindividual scans into the same final global coordinate system.

Laser Scanner

Laser scanning employs triangulation, and a source and detec-tor to accomplish a displacement measurement. A commerciallaser scanner (Keyence LJ-V7300), which comprises of a bluesemi-conductor laser source, and a CMOS detector (Fig. 3)[24], is utilized in this work. The detected signal is obtainedto determine the relative distance to the target. The laser scan-ners working range is 155 mm to 445 mm away from laserhead. This Keyence line laser can obtain the detected signalover its scanned cross section at once, with a resolution over

Fig. 2 Non-contact geometricimperfection measurement rig forcold-formed steel memberspicture and rendering, includingthe following details (a) laserscanner; (b) large dimensionrotary stage; (c) linear stage andsupport frame; (d) precision railsfor linear stage and linear motionsystem

Exp Mech

its cross section of 0.3 mm. The line laser scanning width de-pends on the depth of scan to the specimen and varies from110 mm to 240 mm as shown in Fig. 3; there are 800 points atthe largest width.

The accuracy of the laser depends on the distance to thetarget both vertically and horizontally as shown in Fig. 3.Highest accuracy, 0.1 mm (0.005 in.), is achieved when thetarget is 155 mm to 300 mm away from the laser head. Sincecold-formed steel specimens are highly reflective, extra caremust be taken with stray reflection to reach the target accuracylevels. An embedded double polarization program is appliedto eliminate stray reflection, and the scanning environment iskept dark to remove the influence of ambient light. The laser

head is able to profile the target at high frequencies (up to16 kHz), which allows it to continuously scan in the longitu-dinal direction, as discussed in section 2.4.

Rotary Stage System

The rotary stage system, as shown in Fig. 4, is intended to rotatethe laser head to any desired angle. The rotary stage systemconsists of: a large diameter rotary ring, a timing belt affixedto the rotary ring, and a stepper motor (LongMotor 23HS9430).The rotary ring, as shown in 4a, has a diameter of 635 mm(25 in.). The timing belt is in 32 pitches with 800 teeth in total.A spur gear, with 17.48 mm (0.688 in.) pitch diameter, ismounted on the shaft of the stepper motor and grips the timingbelt. The rotary ring is made of cast iron, which is heavy due toits dimension and density. A rotary ring with double rails(Fig. 4) is selected for the rotary system. The v-shape rails bettergrip the roller bearings that hold the rotary ring and transmit theheavy loads to the support frame. The double rails can effective-ly prevent out-of-plane vibration when the rotary ring is driven.

Stage Support Frame

A triangular support frame is used to hold the rotary stage, anddriven by the linear motion system. The triangular design ofthe support frame prevents the rotary stage and its heavy ro-tary ring from tipping over (Fig. 5(a)). The triangular frameeffectively distributes the load to the rails and provides stablemotion of the laser and rotary system.

Linear Stage System

The linear stage system consists of a motion system (Fig. 5(b))and a linear guide system (Fig. 5(c)). The linear motion sys-tem, as shown in the Fig. 5(b), has a structure consisting of

Fig. 3 Measurement accuracy and scan range for Keyence LJ-V7300,adapted from (note F.S. indicates full scale accuracy which is 0.09 mm atd=300 mm)

Fig. 4 Detail of steel ringemployed for rotary laser stage

Exp Mech

two encoders at the two ends, and the drive chain and sprocketstay in between. Encoder 2 is embedded inside the motor(GM9236S019) system. Two shaft couplers are used to con-nect the motor, the drive chain and sprocket, and Encoder 1(ENS1J-B28). The signal of Encoder 1 is used as the motioncontrol feedback to control the motor, while the signal of

Encoder 2 (HEDS-91X0) is used as the laser position feed-back to trigger the laser scanner. The linear guide system isdesigned to provide smooth motion with double rails and fourball bearing supports. The triangular support frame sits on thefour ball bearing supports. The support rails have a roundnessof 0.00203 mm (0.00008 in.) and straightness of 0.0254 mm(0.001 in.). The four ball bearing supports can self adjustdeviation of motion when the heavy infrastructure movesalong the rails.

Motion Control of Stage

The automation of the geometric measurement platform isrealized by a LabVIEW program through the control sys-tem depicted in Fig. 6. A data acquisition (DAQ) system(NI 6343) connecting to a PC is the center of the controlsystem. The DAQ board connects to a stepper motor driver(M542) and a DC motor driver (MODEL 4212), whichsends a pulse signal and an analog control signal, respec-tively. The stepper motor controller converts the pulse sig-nal to a stepper motor control signal, to power the steppermotor, which rotates the rotary ring to a desired position.The DC motor driver converts the analog control signal toa large power signal to drive the DC motor; the speed ofthe DC motor is changed with the signal. The rotationangle of the DC motor, which indicates the longitudinalposition of the laser scanner, is fed back to the DAQ boardand the laser scanner controller through Encoder 2 andEncoder 1, respectively. The signal from Encoder 1 is usedfor linear motion real-time control, and that from Encoder2 is used for controlling laser measurements. Pulses fromEncoder 2 can switch the laser on and off. When the rising

Fig. 5 Detail of linear stage system (a) stage support frame (b) linearmotion system (c) linear guide system

Fig. 6 Overall block diagram of imperfection measurement rig

Exp Mech

edge of a pulse is detected, the laser is triggered and a laserline is projected on a scanned surface. Measurements of across section where a laser line is projected and reflected tothe detector of the laser head are collected by the lasercontroller and sent to the computer. With the calibrationof the encoder-triggered laser scanning, the distance be-tween adjacent profiles of measurement from the laser isknown, so the longitudinal position of the target specimencan be calculated. Details are provided in Section 4.

Linear Stage Control

The linear motion control aims for precise fixed position con-trol (Fig. 7). The target position is input from the controlprogram in the PC. The DAQ board receives the commandfrom the PC, and converts the command to the analog signalfor the DC motor driver. The DC motor driver then controlsmotion of the DCmotor. Themotion from the DCmotor is fedback to the control program through Encoder 1 and the DAQboard. Since the feedback signal from the encoder is in asquare-wave format, which is counted by the counter terminalon the DAQ board, a factor is applied to the feedback signal toconvert counts to the longitudinal position of the laser scanner.Therefore, the difference Δd between the target position andthe feedback position can be calculated. A function of Δd isused as the command to control the DC motor. The differenceΔd is first converted to error e by a constant empirical param-eter K1 (K1=9.73), and then converted to the control signal dvby the function K2(e). Since there is strong nonlinearity of thedynamics of the DC motor, a linear function for K2(e) is notenough to control the system. A sliding mode control there-fore is used. The function K2(e) is a piecewise function of e(Fig. 8). Error for this final system is addressed inSection 4.3.3.

Rotary Stage Control

Similar to the linear motion control, the rotary stage con-trol is a precise fixed position; however, the control law forthe rotary stage is relatively simpler. The target angle isassigned from the control program in the PC, and sent tothe stepper motor driver. The driver sends the correspond-ing control signal to the stepper motor. This enables thestepper motor to go until reaching a certain position. Cal-ibration is carried out to find the parameter that convertsthe number of steps to the angular position of the laserscanner mounted on the rotary ring.

Calibration and Registration of 3D Points

Linear Stage Calibration

Linear position of the laser system is determined through twoencoders attached to the drive shaft of the liner motion systemas shown in Fig. 5(b). As depicted in structure of Fig. 6 En-coder 1 provides the primary feedback to the linear motionsystem and Encoder 2 provides triggering for the laser. Thelinear motion calibration is aided by two measurement stan-dards: a caliper with a range up to 635 mm (25 in.) and anaccuracy of 0.127 mm (0.005 in.), and a digital dial gaugewith a range up to 25 mm (1 in.) and an accuracy of0.0127 mm (0.0005 in.). The standards are installed alongthe linear rails as shown in Fig. 9 and allow for precise mea-surement of the triangular support frame over two distinctdistance ranges.

Fig. 8 Controlling linear position error, function of K2

Fig. 9 Calibration of linear motion system (a) digital caliper for >10 mmmovement (b) dial gauge for <10 mmFig. 7 Simplified block diagram of linear motion controller

Exp Mech

Encoder 1 calibration

Encoder 1 provides the primary longitudinal measurement forthe linear motion system. Rotation count on the encoder isconverted to displacement along the linear rails through cali-bration with the caliper and dial gauge measurement standard(Fig. 9(a) & (b)) and the results are provided in Fig. 10(a).Measured several times, the response is linear and the linearcalibration coefficient is subsequently used for the measure-ment platform.

Encoder 2 calibration

Encoder 2 is used to trigger the laser so that measure-ments occur at discrete intervals along the length even ifthe velocity/time of scanning has some variation. In addi-tion, Encoder 2 also provides, through calibration, a mea-surement of longitudinal laser location. Rotation count on

the encoder is converted to displacement along the linearrails through calibration with the caliper measurementstandard (Fig. 9(a)) and the results are provided inFig. 10(b). Similar to Encoder 1 the movements are re-peated several times, the response is linear, and the linearcalibration coefficient is subsequently used for the mea-surement platform.

Rotary Stage Calibration

Calibration of the stepper motor, which provides rotationof the rotary ring, is completed using a simple inclinom-eter clamped directly to the laser head mount as shown inFig. 11. The inclinometer has an accuracy of 0.01°. Aseries of rotation trials are conducted and the counts fromthe stepper motor are plotted against the rotation angle inFig. 12(a). A linear calibration coefficient is establishedby regression and subsequently employed for control ofthe rotary ring.

Motion of the rotary stage system is imperfect. Therotary ring is heavy, the timing belt (Fig. 4) is plastic,and there is some friction in the lateral guides. This leadto some concern regarding repeatability and hysteresis inthe rotary system. Therefore, accuracy between commandand actual rotation of the rotary stage is studied beyondthe initial calibration. Starting always from 0° (vertical)angles between −140° and 140° are studied at 10° incre-ments and the difference between desired and actual rota-tion are recorded and provided in Fig. 12(b). The maxi-mum observed error is 0.2° and the mean error from allrotations is 0.04°.

Registration of Points in Global 3D Space

3D registration framework

The overall goal of the device is to register the surfacecoordinates for the scanned cold-formed steel member.

Fig. 10 Linear calibration factors and results for encoders (a) Encoder 1scale factor and fit (b) Encoder 2 scale factor and fit

Fig. 11 Calibration setup (with clamped digital angle) for rotary motionsystem

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Each triggered scan from the line laser provides 800 read-ings in a local coordinate system located a distance d (d=300 mm) from the laser head. This local reading must beconverted into global coordinates so that multiple read-ings may be combined to provide the complete scan. Ig-noring initially the longitudinal location (y) and dealingwith a specific cross-section (X-Z plane), Fig. 13 providesthe coordinate systems and notation used to make thisbasic transformation from local to global coordinates forthe laser measurement.

The initial local coordinates x 0, z 0 are defined by theorigin of the laser readings at an angle of 0°. The z0 axisof the local coordinate system is vertical and perpendicu-lar to horizontal ground. The global coordinates X , Z aretranslated from the initial local coordinates following avector m0 so that the origin of the global coordinates, O,

is located at the center of the rotary ring. Note, the lineO0A0 (d=300 mm) that passes through the origin of thelocal coordinates does not coincide with the radius OA0

(R=281.2 mm) that passes through the center of the rotaryring, and thus there is a constant distance OO0 (r=18.2 mm) between the origin of the laser local coordinatesand the center of the rotary ring.

As the laser rotates to any angle θ, the center line of thelaser OiAi also rotates θ along the circumferential axis of therotary ring. This results in radius OAi and a distance OOi

between the origin of the new ith laser local coordinates andthe center of the rotary ring. The angle between OAi and OiAi

is the same as that between OA0 and O0A0. The radius of therotary is constant so that OA0 equals OAi. The distance (OOi)and the angle between the origin of laser local coordinates andthe center of rotary ring in position 2 equals the distance (OO0)and angle in position 1.

To register all local measurements into global coordi-nates a fixed reference point P is established. In theinitial coordinates x0,z0 the fixed reference point P ismeasured, and the position vector p0 is established. Thedistance between local coordinates origin to the rotaryring center is m0, which is unknown. The free vectorn0 from O to P must follow:

m0 þ n0 ¼ p0 ð1Þ

For the laser at any non-zero angle θi, the fixed referencepoint P is measured in the new local coordinates xi,zi and anew position vector pi is obtained. The position vector mi forthe rotary ring center O should have the same vector expres-sion as m0 since the two vectors both deviate an angle α from

Fig. 12 Rotary stage calibration results (a) scale factor and fit in terms ofmotor steps (b) rotary stage accuracy in °

Fig. 13 Global coordinates (X,Z) and laser local coordinates (x,z) fordifferent measurements (note, only single point laser at center of linelaser illustrated for clarity)

Exp Mech

the z0 axis and zi axis and are both equal in length to thedistance r. Thus:

m0 ¼ mi ¼ m ð2Þ

Further, similar to Eq. (1):

mi þ ni ¼ pi ð3Þ

Noting, by simple rotation:

ni ¼ Tin0 ð4Þ

where Ti ¼ cos θið Þ sin θið Þ−sin θið Þ cos θið Þ� �

Substituting eq. (4) into eq. (3) and recognizing the equalityof eq. (2):

mþ Tini ¼ pi; i ¼ 0; 1; 2;… ð5Þ

Considering the measurement noise in an actual experi-ment, we may define the error between the independentlydetermined pi and that determined through the transformationas:

ei ¼ pi−m−Tini ð6Þ

E ¼X

eTi ei ð7Þ

By minimizing eq. (7), m may be found such that the rela-tionship between O0 and O is identified.

With m known, a laser reading (Fig. 14) in local co-

ordinates (li )(i.e., xi, zi), may be translated to the rotaryring center coordinates by adding m and then rotating tothe final global coordinate system with Ti, i.e., a laserreading in the global coordinate system, Li (i.e., Xi, Zi),is as follows:

Li ¼ Ti li−m� �

ð8Þ

3D registration calibration

A calibration set up is designed to experimentally find mper the procedure of the previous section. As shown inFig. 15 a 0.0625 mm (1/16 in.) thin string is tied to twoclamps that grip to a support beam to establish the refer-ence point. The reference point is scanned at 25 angles (0°)to ±120° with an increment of ±10° and at 8 locationsalong the length. From this data and employing eqs.

(5)-(7) the position vector m is estimated to be: m ¼6:328; 17:016ð ÞT (mm). With m established the local toglobal transformation is provided through eq. (8). Themeasurements at any corresponding angle θ can be trans-formed and translated; following the registration matrix, tothe global coordinates so that different measurement sur-faces can be registered in the global coordinates.

Validation of calibration model and evaluation of registrationestimation

The global registration leading to eq. (8) assumes therelative position vector m between origins of the laserlocal coordinates and the center of the rotary ring isconstant. The validity of this assumption is tested by astatistical hypothesis test using analysis of variance(ANOVA).

Fig. 14 Global coordinates Li (Xi, Zi) and laser local coordinates li (xi, zi)for different measurements Fig. 15 Calibration setup for 3D registration study

Exp Mech

Validation of calibration model

Specifically, 25 different angles are used to estimate m(i.e., m̂i ) at a given location. This estimation is performedfor 7 different independent locations of the reference point,p. Thus we postulate that the null hypothesis, H0: mj ¼ mfor all j=1,2,…,7. The F-statistic can be formed by takingF(6168). At a 95 % confidence level if the calculated F

values are smaller than 2.15, then H0 is true. Since m ¼x; zð ÞT is a vector, two F values, 0.428 and 1.006, are cal-culated corresponding to x and z respectively. Both valuesare smaller than the critical F value, thus the null hypoth-esis is accepted and a constant m is justified by theobservations.

Evaluation of registration estimation

The total variance in m can be estimated as follows:

var mð Þ ¼ SSTDFT

¼

X7j

X25i

m ji−m� �2

25� 7ð Þ−1 ¼ σ2X ;σ

2Z

� �T¼ 0:2022; 0:0819ð ÞT ð9Þ

Where σX=0.45mm and σZ=0.29mm. The accuracy of re-construction model is:

e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2

Xþ σ2

Z

q¼ 0:533mm ð10Þ

Considering the radius (R=281.2 mm) of the rotary ring,this deviation of the rotary center estimation is small. Theestimation therefore is satisfactory and good for initial surfaceregistration.

Example – Scanning a z-Shaped Cold-Formed SteelPurlin

To provide a brief example, a z-shaped cold formed steelspecimen as commonly employed in metal building roofsand walls, and currently being investigated in our struc-tural testing lab was scanned. The specimen was magnet-ically mounted to the support beam of the scanning rigas shown in Fig. 16(a). Note, the specimen has two thicksteel plates welded to its ends, necessary for subsequentstructural testing.

�Fig. 16 Examples scan of Z-shaped cold-formed steel specimen (a)Z-shaped cold-formed steel specimen (with welded-on end plates fortesting) (b) raw scan at 0° (c) raw scan at −80° (d) raw scan at 100°

Exp Mech

Longitudinal scans are performed at angles of 0°, −80° and100° as provided in Fig. 16(b)-(d). The specimen is approxi-mately 250 mm long. Encoder 2 provides fixed triggering ofthe laser scans in the longitudinal direction. Therefore, allthree scans have mutual longitudinal readings at a spacing of3.11 mm. The local x data is known based on the initial posi-tion across the head of the laser and the local z data is the directoutput of the laser.

Per eq. (8) and the registration data m (6.368,17.016),the local coordinate information is explicitly transferredto global coordinates. Specifically, local coordinates x,zare transferred to global coordinates X,Z, at 0°, −80° and100° via:

X ið Þ

Z ið Þ

¼ 1 0

0 1

� �x ið Þ0

z ið Þ0

( )− 6:368

17:016

!ð11Þ

X ið Þ1

Z ið Þ1

( )¼ cos −80∘ð Þ −sin −80∘ð Þ

sin −80∘ð Þ cos −80∘ð Þ� �

x1ið Þ

z1ið Þ

− 6:368

17:016

� �

ð12ÞX ið Þ

2

Z ið Þ2

( )¼ cos 100∘ð Þ −sin 100∘ð Þ

sin 100∘ð Þ cos 100∘ð Þ� �

x2ið Þ

z2ið Þ

− 6:368

17:016

� �

ð13Þ

Once the data is combined into a common coordinatesystem it may be readily plotted. For the visualization ofFig. 17 the image is colored based on the deviation fromthe average straight and level surface location referencedfrom the mid-plane. Overall a twist in the section andwaviness in the plates may be observed. Additionalpost-processing of dimensions and other imperfectionproperties is possible. In addition, this geometry maybe used as the basis for subsequent finite element modelsor for other forms of analysis.

Conclusions

This paper details the design of a measurement platform forscanning the imperfect geometry of relatively large parts withhigh precision using a line laser mounted on translational androtary stages. The intended application is the geometric mea-surement of cold-formed steel members as used in the con-struction of buildings. The mechanical realization of the plat-form is demonstrated in the paper by detailing its three majorcomponents: laser scanner, linear motion system, and rotarystage system. Control and calibration for each component isprovided. The platform can accurately scan a part up to 3 m inlength and 250 mm in diameter as configured. Multiple scansare required to build-up the geometry—data of surfaces mea-sured from different angles of view is transformed from localcoordinates associated with the laser scanner to global coor-dinates, a process requiring calibration as detailed herein. Amulti-pass scan of a z-shaped cold-formed steel member isprovided as an example to demonstrate the efficacy of themeasurement platform. The point cloud of the part providesdata for performing additional analysis on dimensions, plateimperfections, etc., and can form the geometric basis for directfinite element simulations.

Acknowledgments We would like to thank Nick Logvinovsky for allhis help in designing and building the experimental platform at JohnsHopkins University. This work was partially funded by the NationalScience Foundation (NSF), Grant 1235196. Any opinions, findings, con-clusions, or recommendations stated are those of the author(s) and do notnecessarily reflect the views of the National Science Foundation.

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